I would suggest that you should not use Watergate or the Iran-Contra Affair as reasons for the change towards conservatism. Both were done by conservative presidents and would not really have made people more likely to support conservative ideas.
To explain this change, look to the 1960s. This was a decade of huge changes. There was the Civil Rights Movement, the Vietnam War, the Great Society, and the counterculture. All of these things pushed the US towards more conservatism.
In the 1960s, the changes caused many Americans from Nixon's "silent majority" to feel that the US was changing too fast in cultural terms. The protests against the war and for minority rights worried traditionalists and caused a backlash. The spending on the war and the Great Society hurt the government finances and made people more conservative in terms of economics. The move towards conservatism was a backlash against the liberalism of the 1960s.
You can argue that Reagan's election helped to push the move along since he became a popular conservative president. But be sure to note that it was also a result of the changes. Reagan would not have been elected if the US had not already been moving towards conservatism.